On the $W$-action on $B$-sheets in positive characteristic

Abstract:

Let $G$ be a connected reductive group defined over an algebraically closed base field of characteristic $p\ge 0$, let $B\subseteq G$ be a Borel subgroup, and let $X$ be a $G$-variety. We denote the (finite) set of closed $B$-invariant irreducible subvarieties of $X$ that are of maximal complexity by $\mathfrak {B}_{0}(X)$. The first named author has shown that for $p=0$ there is a natural action of the Weyl group $W$ on $\mathfrak {B}_{0}(X)$ and conjectured that the same construction yields a $W$-action whenever $p\ne 2$. In the present paper, we prove this conjecture.